# Math Golf Project

### By Swetha Tandri and Shravya Mahesh(2nd period)

The height h(in feet) above the ground depends on the time, t(in seconds) it has been in the air. Earl hits a shot off the tee that has a height modeled by the function f(t)= -16t^2+100t. Look below to see how we evaluate his shot with many different aspects. It is time for us to perform another "mathvestigation!"

## Question #1

Graph this Function= f(t)=-16t^2+100t
This graph has an axis of symmetry of 3.13 seconds and the maximum value of the parabola is 156.25 feet. This makes the vertex of (3.13, 156.25). The roots of this function are (0,0) and (6.25,0). The total amount of seconds the ball was in the air was 6.25 seconds.

Function Domain: All Real Numbers

Function Range: y is equal to or less than 156.25

## Question #2

What are the independent and dependent variables in this situation?

Independent Variable: time in seconds

Dependent Variable: height in feet

The independent variable is the x value and the dependent variable is the y value. The seconds is the value of x on the function while f(x) or y represents the distance.

The height of the ball depends on the amount of time it takes. The time is on the X-axis because it is being manipulated, and the height is the y-axis since it is being changed by the time. If the time is less, the height will change.

## Question #3

What is a reasonable domain and range for this function?

Domain: all real numbers

Range: all numbers less than or equal to 156.25

The domain is all the X-values. Since the graph is a parabola, the lines keep spreading downwards, so the line will eventually cover every X-value. Hence, the domain is all real numbers. However, since the range is all y-values and there is a maximum height, the range must be all values less than the maximum height, or 154.

## Question #4

How long is the golf ball in the air?

The golf ball is in the air for 6.25 seconds. That is the amount of time it takes for the ball to reach the ground again. We know this by looking at the roots of the parabola. The first root goes through the origin and the second root goes through (6.25,0). We use the roots to find the distance because they represent the x value(seconds) when the y value(distance in feet) is zero. Just subtract the values to find that the difference of the zeroes, which is 6.25 seconds.

## Question #5

What is the maximum height of the ball?

The maximum height of the ball is 156.25 feet. That is the highest point of the parabola, also known as the axis of symmetry. This is the maximum height because, on the graph below, there is no higher y value than 156.25. The graph curves downward after that value. The range also specifies that all y values equal to or less than 156.25 would work for the function. That is why 156.25 feet is the maximum height.

## Question #6

How long after it is hit does the ball reach the maximum height?

As shown on the graph below, the X-axis of the vertex is 3.13 seconds, which is how long it takes the ball to reach maximum height. That is also the axis of symmetry. After this value, the x vales keep increasing but the y values decrease because of domain and range.

The graph above shows the vertex of the function, which gives the maximum height, and the seconds needed to reach that height. (3.13 seconds)

## Question #7

What is the height of the ball at 3.5 seconds? Is there another time at which the ball is at the same height? If so, when?

At 3.5 seconds, the ball is at 154 feet. The ball is also at the same height at 2.75 seconds. This is because on a parabola, since the highest point goes down in both ways, the same height will have two different times, as shown below. This is because if you have the axis of symmetry as an imaginary line, both sides of the parabola would look the same giving each y value 2 x values. One of the coordinate pairs would be on 1 side of the vertex and the other pair would be on the other side. The graphs below show the two x values for the y value of 154 feet.

## Question #8

At approximately what time is the ball 65 feet in the air? Explain.

The ball would be approximately 65 feet in the air at 0.74 seconds and also at 5.51 seconds because of the results of the the graphs below. The reason that 65 feet has two results is because the function is a parabola and it is U shaped. This causes there to be two x values for every y value except the vertex because that is the point on the top of the graph before it goes down again. One coordinate pair(5.51,65) is on on one side of the vertex, and the other pair(0.74,65) is on the other side. The graphs below are not completely exact, but we rounded our results to the nearest hundredth.

## Question #9

Tweety Bird takes off from the green at the same time you tee off. His height is increasing at a rate of 4 feet per second. When will he be at the same height as your ball? What is the height? Graph the scenario.

The Blue Parabola is Earl's Ball.

The Green Line is Tweety Bird's Flight.

The function of Tweety Bird's flight is y=4x. In the question, it specified that his flight is linear so he travels in a straight line. He takes flight from the ground, so there is no y intercept. The function also represents Tweety Bird's flight of 4 feet per second mentioned in the question. Below is the graph of the ball and the bird. They intersect at the point (6,24). The x value represents the time, which is in seconds, and the y value is the distance in feet. So tweety bird met the golf ball in 6 seconds at 24 feet.

## Question #10

Suppose Gloria and Earl stand side by side and teed off at the same time. The height of Gloria's ball is modeled by the function f(t)=-16t^2+80t. Earl hits a shot off the tee that has a height modeled by the function f(t)=-16t^2+100t. Whose golf ball will hit the ground first? How much sooner does it hit the ground? How high will Gloria's ball go? Compare the two shots graphically.

The Blue Parabola is Earl's ball.

The Purple Parabola is Gloria's ball.

According to the roots of the graph, Gloria's ball would hit the ground first, at 5 seconds. Earl's ball lands 1.25 seconds later because his function had a 20 units greater B value than Gloria. The maximum value for Gloria's ball is 100 feet, which it reached in 2.5 seconds. Compared to Earl's ball, Gloria's ball was 56.25 feet lower. Another aspect that we could compare from the graphs is how long the balls were in the air. Gloria's ball was in the air for 5 seconds unlike Earl's ball that lasted for 6.25 seconds. One thing that the graphs have in common is the shape of the parabola. Since both the absolute values of "a" are greater than 1, the graphs are narrow.

## Question #11

Suppose that Earl hit a second ball from a tee that was elevated 20 feet above the fairway.

Graph of Ball:

## #11a

What effect would the change in elevation have on the graph?

One effect the change in elevation would have on the graph, is that it would have a y intercept, which was not present in the old function. Since y represents the height, the new graph shows that the ball started 20 feet high in zero seconds, before he hit the ball. The y intercept would also affect the maximum height and how long it would take to reach there. On the previous function, it took 3.13 seconds for the ball to reach 156.25 feet. With the new positioning of the ball, the maximum height of the ball is 176.25 feet in 3.13 seconds, 20 feet higher. Lastly, since the ball has a higher vertex, it would stay in the air for 0.19 seconds longer.

## #11b

Write a function that describes the new path of the ball.

f(t)= -16t^2+100t+20

What is different about this function than the old one is that the c value of the function is 20 instead of 0. In a quadratic function, the c value determines the translation of the parabola. The ball started on a tee 20 feet higher than the fairway, so the parabola should be shifted 20 feet higher. The remaining parts of the function are the same.

## #11c

Graph the new relationship between height and time. Make sure to label the graph and to graph the original function as well as the new function in the given graph.

As you can see in the graph, they have the same axis of symmetry, which is 3.13 seconds. The vertex of the new function is 20 feet higher than the original function because the new function has a y intercept of 20. The graphs have the same shape because the a values are the same. Overall, the elevation of the graph made the ball go higher and the ball stayed in the air longer by 0.19 seconds.

## #11d

What would be a reasonable domain and range of this new function?

Domain: All Real Numbers

Since this is a parabola and goes infinitely in one direction, the domain would always be all real numbers.

Range: y is equal to or less than 176.25 feet.

The range would be different from the original function because the vertex changed in the new function. The y value of the vertex is 176.25, the maximum value. All y values would be equal to or less than the maximum value.